Strengthening Hadwiger's conjecture for 4- and 5-chromatic graphs

Abstract

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [11] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and $S\subseteq V\left(G\right)$ takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S.

We prove this conjecture in the first open case of $t=4$. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows:

Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph.